Sunday, April 26, 2009

The answer to the question in the post title is obvious to a certain extent. Mainstream economists are part of a larger system that is served by obfuscation. If (some) economists become too penetrating in their examination of a capitalist economy, their colleagues simply purge them from the profession. (I seem to recall reading somewhere that Fred Lee is indeed writing a book.)

"Marx believed that capital is not a complement to but a substitute for labor. Thus technological progress and capital accumulation that raise average labor productivity also lower the working-class wage. Hence the market system simply could not deliver a good or half-good society but only a combination of obscene luxury and mass poverty. This is an empirical question. Marx's belief seems to me to be simply wrong." -- Brad DeLong

I see here an incorrect belief that income can be determined by technical relationships. Given DeLong's reference to complements and substitutes, he is, I think, drawing on his (mis)understanding of marginalism. He probably believes something like the following: in a market economy, agents receive (or should receive) the value of the marginal products of the services of the factors or production that they own. (By mentioning ownership, I am already being more careful than many might be; capitalists do not have a marginal product, even if the services of capital goods were to each have one.) To reinforce my point that some believe this, I quote some comment on a blog somewhere:

"And then you can have a simple theory of exploitation where it happens if somebody somewhere doesn't get their marginal product." -- Radek

I havepointedoutbefore that marginal productivity, when correctly stated, is a theory of the choice of technique, not a theory of distribution. And, by quoting Duncan Foley, I acknowledged the unoriginality of my understanding. I now take the opportunity of my recent exposition of a reswitching example to re-iterate the non-existence of the marginal productivity theory of distribution. Figure 1 illustrates that example, and it is consistent with all correctly formulated conditions on marginal products for the example.

Figure 1: Wage-Rate Of Profits Curves

I want to consider what data about production the observing economist must know to calculate the value of marginal products. First, suppose he knows the technique in use. In each (non-vertically integrated) industry, the inputs purchased by each firm and the outputs are known. In a circulating capital model of competitive markets, this data determines wage-rate of profits surfaces, of which two are shown in the figure. But the observing economist cannot use this data to determine the equilibrium distribution; any point on the surface corresponding to the observed technique is consistent with the data.

But marginal products are defined in terms of counterfactual experiments. Accordingly, suppose an economist knows all available production processes, as well as the technique in use. One can use this data to construct the wage-rate of profits frontier, which is the outer envelope of the wage-rate of profits curves constructed for each technique. For the sake of argument, suppose an uncountably infinite number of techniques happen to exist, and these techniques vary continuously along the frontier. So the frontier contains no switch points, where more than one technique is cost minimizing. Nevertheless, if the rate of profits (or wages) varies an infinitesimal amount from the value corresponding to a location on the frontier, a different technique would be cost-minimizing in a long-run equilibrium.

Reswitching shows this data is not necessarily enough to determine distribution. In the example, the alpha technique corresponds to two discrete ranges of equilibrium wages. It may be that if wages were set to a completely different level, firms would want to employ the same number of people, purchase the same resources, and use the same processes in production. The quantity flows within the firms would be unchanged, although who would purchase what on consumption markets could be vastly different. And yet for some level of wages between these two values, firms would want to adopt other processes. Thus, one's income in a capitalist economy cannot be a reward for physical productivity. The belief that income rewards productivity in a capitalist economy is an element of vulgar political economy, contradicted by rigorous price theory.

"...one who believes technology to be ... like my 1966 reswitching example ... will have a more sanguine view about how successful militant power by organized labor can be in causing egalitarian shifts in the distribution of income away from property even in the long run." -- Paul A. Samuelson, "Steady-State and Transient Relations: A Reply on Reswitching", Quarterly Journal of Economics, V. 89, N. 1 (Feb. 1975): 46

Suppose you run into an economist who refers to the marginal product (or the value of the marginal product) of some factor of production, perhaps in explaining why, regrettably, wages for somebody cannot easily be made higher. That economist simply does not know what he is talking about.

Thursday, April 23, 2009

When I think of P. J. Proudhon, I think of two works: What is Property? and The Philosophy of Poverty. Apparently, System of Economical Contradictions; or, the Philosophy of Misery is a translation of the latter's full title. And I "know", from Karl Marx's The Poverty of Philosophy, it is a very badly argued book.

The Project Gutenberg collection for Proudhon consists of exactly those two books. The formats plain text, sometimes html, Mobi pocket, and other protable formats, but no PDF.

Shawn Wilbur recently announced annouced the New Proudhon Library. I found the organization of the wiki confusing. This page has external links, but not to the Project Gutenberg texts. I guess this page has links to the original French.

Sunday, April 19, 2009

1.0 IntroductionThis is another example of reswitching and capital reversing. In this example, a capital good can only be used with a fixed quantity of labor in producing the single consumption good. If entrepreneurs would like to employ a different quantity of labor per unit output of the consumption good, they must use the services of some other capital good. In other words, this is an example of fixed coefficients, in some sense. But the existence of reswitching and capital-reversing are compatible with variable coefficients and the possibility of marginal adjustments. I think many economists may be confused on this point.

This example can also be seen as an illustration of a generalization of Paul Samuelson's "Surrogate Production Function" model. Samuelson assumed that alternative processes for producing the consumption good each require the use of a different capital good. But Samuelson required the special-case assumption that, for a given capital good, the desired ratio of physical units of the capital good to labor was invariant among processes producing the consumption good and producing the capital good. As Garegnani pointed out, this special case effectively collapses Samuelson's model to a one-good model. And so Samuelson's attempted defense of the aggregate production function fails.

2.0 The Technology and Quantity FlowsConsider a firm whose managers are aware of the technology shown in Table 1. Each column defines a Constant-Returns to Scale process for producing the output indicated by the column heading. The managers know of two processes for producing corn, a good used only in consumption. Steel and labor are each used in producing either more steel or in producing corn. Similarly for tin. Each process requires a year to complete, and the capital good is totally used up in producing the output. In the jargon, this is an example with no fixed capital; all capital is circulating capital.

Table 1: The CRS Technology

Inputs

SteelIndustry

TinIndustry

Corn Industry

Alpha

Beta

Labor (Person-Years):

1/3

1/2

1

3/2

Steel (Tons):

1/6

0

1

0

Tin (Tons):

0

1/4

0

1/4

Corn (Bushels):

0

0

0

0

Output (Various):

1

1

1

1

Two techniques are available for the vertically-integrated firm to produce corn. Each technique consists of two processes operated in parallel. The first technique, called the alpha technique, consists of the steel-producing process and the first of the two corn-producing processes. Suppose these processes are operated at a scale to produce 6/5 tons of steel gross and one bushel corn. Of the output of the steel-producing process, 1/5 tons replaces the steel inputs used up in producing the steel. The remaing ton replaces the steel used up in producing the corn. Thus, at this scale, the net output of the vertically-integrated firm consists of one bushel corn. Since 7/5 person-years are used across both processes, the ratio of the capital-inputs in physical terms to worker is 6/7 tons steel per person-year. Output per worker consists of 5/7 bushels per person-year.

The second technique, called the beta technique, consists of the tin-producing process and the second corn-producing process. When these processes are used to produce 1/3 tons tin gross and one bushel corn, the net output consists of one bushel corn. At this scale of operations, 5/3 person-years are employed. The ratio of the capital-inputs to worker is 1/5 tons tin per person-year. Output per worker is 3/5 bushels per person-year.

The alpha technique produces more output per worker. In the traditional and incorrect neoclassical analysis of the production function, greater output per worker for a known technology is achieved by the use of more capital per worker. Firms choose to use a less labor-intensive process, in this incorrect analysis, when consumers choose to save more and the interest rate (called the rate of profits below) consequently falls. Notice that one cannot tell from the above arithmetic with the example whether the capital-to-labor ratio is higher in the alpha or in the beta technique. The units of comparison so far are incommensurable.

3.1 The Alpha TechniqueIf the alpha technique is cost-minimizing, the following equations must hold:

(1/6) ps (1 + r) + (1/3) w = ps

ps (1 + r) + w = 1

where ps is the price of steel in units of bushels corn per ton steel, w is the wage in units of bushels corn per person year, and r is the rate of profits. These equations incorporate the assumption that wages are paid out of the surplus at the end of the year. This system of equations shows that the same rate of profits is earned in producing both steel and corn.

This is a system of two equations in three variables. As Sraffa notes, it has one degree of freedom. Two of the variables can be found in terms of the third. I take the rate of profits as the independent variable. The wage is then:

wα(r) = (5 - r)/(7 + r)

And the price of steel is:

ps(r) = 2/(7 + r)

The wage-rate of profits curve is a declining function in the first quadrant of the wage-rate of profits space (Figure 1). Since this is a model of the production of commodities by means of commodities, the wage-rate of profits curve cuts both axes. The maximum rate of profits in the alpha system, found when the workers live on air, is r = 500%. The maximum wage, where the capitalists receive none of the surplus, is 5/7 bushels per person-year.

The price of steel can be used to evaluate the capital goods used per worker in the alpha technique:

Iα(r) = 12/[7 (7 + r)] bushels per person-year

The price Wicksell effect is the variation, given the technique, in the value of capital per worker with the rate of profits.

where pt is the price of tin in units of bushels corn per ton tin. The solution of this system of equations is:

wβ(r) = (3 - r)/(5 - r)

pt(r) = 2/(5 - r)

The maximum rate of profits in the beta system is r = 300%, which is lower than the maximum in the alpha system. The maximum wage in the beta system is 3/5 bushels per person-year, which is also lower than in the alpha system. The value of capital per worker is:

Iβ(r) = 2/[5 (5 - r)] bushels per person-year

3.3 Switch PointsThe technique with the highest wage, at a given rate of profits, is the cost-minimizing technique at that rate of profits. The wage-rate of profits frontier in models like the example is the outer envelope in Figue 1 of all wage-rate of profits curves. The alpha technique is cost-minimizing both for rates of profits between 0% and 100% and for rates of profits between 200% and 500%. The beta technique is cost minimizing for rates of profits between 100% and 200%.

At a switch point, more than one technique is cost minimizing. In the example, switch points are at (r, w) = (100%, 1/2) and at (200%, 1/3).

3.4 Selected Consequences for Capital and Labor "Markets"The analysis of the choice of technique allows one to graph the value of capital per worker in a steady state versus the rate of profits (Figure 2). The graph displays a combination of price and real Wicksell effects. Consider the horizontal lines at the rate of profits of 100% and 200% for the switch points. The real Wicksell effect is the variation in the in the value of capital per worker with the technique at a rate of profits for a switch point. The price Wicksell effect is shown by the curves not being vertical between switch points. In Figure 2, the switch point with a positive real Wicksell effect is indicated. The example demonstrates the logical invalidity of traditional neoclassical theory, in which real Wicksell effects are always negative.

Figure 2: Investment Function

The analysis of the choice of technique also allows one to graph the level of employment firms offer at each wage, given the level of net output. Figure 2 indicates the switch point around which firms will attempt to hire more labor if the wage is increased.

Figure 3: Labor Employed per Unit Net Output as a Function of Wages

As an aside, I wonder if the following quotation includes an allusion to the Cambridge Capital Controversy:

"For output in goods-producing industries to increase, the quantity of intermediate goods used to produce the output must also be increased (e.g. cloth to produce a shirt, ham to produce a ham sandwich, etc.). However, the concept of the marginal product of labor (or capital) requires that as the input of labor (or capital) is increased, all other inputs must be held constant. But this is not possible for intermediate inputs in goods-producing industries. Therefore, the concept of the marginal product of labor (or capital) is not possible when there are intermediate goods in the production function. Again, we prefer not to teach such logical errors to our students." -- Bernard Guerrien and Emmanuelle Benicourt (2008)

4.0 ConclusionsSo much for the logical validity of typical arguments for improved labor market flexibility and for the belief that minimum wages cause unemployment.

Wednesday, April 15, 2009

J. R. Hicks invented the IS/LM model to compare and contrast his interpretation of Keynes' General Theory with "classical economics". Franco Modigliani later appended a supply and demand model of the labor market. Even later, John Hicks rejected this model.

Offhand, I can think of three issues with the IS/LM model:

The LM curve shows a stock equilibrium, while the IS curve shows a flow equilibrium. Thus, it is not clear to me that they can be graphed on the same diagram.

Both curves are drawn for a given set of expectations. If one curve shifts, the expectations upon which the other is drawn can hardly remain constant. Thus, both curves must shift and the equilibrium point becomes indeterminate.

The curves replicate the separation between nominal and real values that Keynes was trying to transcend with his monetary theory of production. Thus, the model cannot be a valid representation of the theory Keynes was so laboriously trying to express.

The first point challenges the internal validity of the model. The second suggests the model cannot empirically predict. The third is a point about the history of economic thought and would be expressed better if I remembered better some of Keynes post-General Theory work.

Thursday, April 09, 2009

The best understanding of the Cambridge Capital Controversy is still an unresolved question, without a consensus having been reached. This presents an opportunity for fans of any long departed economist writing on capital theory. They can declare him or her the victor in the CCC avant le letter.

I barely recall Santiago Valiente (1980). But I seem to remember somebody declaing Knight the winner of the CCC.

On the other hand, Jack Birner argues that Hayek should be considered the winner:

"If the article [Hayek 1934] had been more widely known and understood, there would never had been a Cambridge debate. [Footnote:] Co. for instance [Hayek 1934], note 2 on pp. 212-13, which explains why there is not always a one-to-one correspondence between the rate of interest and the value of capital. Hayek's highly sophisticated analysis of the relations between input and output through time makes use of the same three-dimensional diagrams that seven years later were to constitute the core of [The Pure Theory of Capital]. Not only is the model of 1934 taken over almost unchanged in PTC, it is used there to analyze the most complicated of the various cases Hayek distinguishes, the one with durable capital goods." - Jack Birner (1999)

Some ironies arise here. Hayek and Knight opposed one another in the second great capital controversy in neoclassical economics. I think Birner is correct in arguing that Hayek thought of his triangles and the analysis in Prices and Production as a simplified and incomplete version of his more advanced capital theory. But that simplified version is itself a rejection of the even simpler Austrian theory associated with Böhm Bawerk and the "average period of production".

Anyways, my refutation of Garrison's version of Austrian Business Cycle Theory has been rejected by the Cambridge Journal of Economics. One of the reviewers stated that I need to rewrite it in view of Birner's article. I intend to take this advice.

So in the process of responding to reviews, I have retreated from arguing that ABCT is mistaken to arguing merely that Garrison's version of ABCT is mistaken. Garrison's version is the most prominent among scholars. I'll leave open whether an internally valid ABCT can possibly be constructed on Hayek's Pure Theory of Capital and "Ricardo effect" analysis. I am hardly alone, however, in doubting that it can be:

"The book [PTC] could not achieve its aim, because of Hayek's lack of formal and mathematical skills and the impossibility of the task itself [Footnote:] as taught by the outcome of the Cambridge capital controversies" -- Hansjörg Klausinger (2006)

"Whether or not Hayek's discussion of the Ricardo effect is refuted by arguments made in the Cambridge capital debates will be left aside in this paper." -- Theodore Burczak (2001: 64)

Sunday, April 05, 2009

"I admit that my criteria of falsifiability does not lead to an unambiguous classification. Indeed, it is impossible to decide, by analyzing its logical form, whether a system of statements is a conventional system of irrefutable implicit definitions, or whether it is a system which is empirical in my sense; that is, a refutable system. Yet this only shows that my criterion of demarcation cannot be applied immediately to a system of statements - a fact I have already pointed out... The question whether a given system should as such be regarded as a conventionalist or an empirical one is therefore misconceived. Only with reference to the methods applied to a theoretical system is it at all possible to ask whether we are dealing with a conventionalist or an empirical theory. The only way to avoid conventionalism is by taking a decision: the decision not to apply its methods. We decide that if our system is threatened we will never save it by any kind of conventionalist stratagem." - Karl Popper (1968): 81-82

John Davis (2009) distinguished between two ways of dividing economists up: based on the content of their theories and based on more sociological criteria of citation networks, conference attendance, professional society membership, textbooks, etc. I think Davis' taxonomy remains of interest even if one does not agree with his views on trend in the economics profession.

Davis distinguishes between orthodox and heterodox economics on the basis of the substances of their theories. In the last column of Table 1, I have listed some distinguishing precepts of orthodox economics. The last three precepts roughly correspond to the opposite of the distinguishing features, according to Davis, of heterodox economics around 1980. I think one could also call orthodox economics "neoclassical". Heterodox economics rejects some combination of the precepts of orthodox economics. For Davis, mainstream economics is a sociological category. The first two columns of Table list some examples. Mainstream heterodox economics may become orthodox in time, with game theory perhaps already having succeeded, at least partially. At any rate, mainstream heterodox economists have access to the leading journals, a presence in the graduate schools generally rated to be the top, and so on.

Table 1: Divisions Among Economists

Non-MainstreamEconomists

Mainstream Economists

Heterodox Economics

OrthodoxEconomics

Marxism

Radical political economy

Institutionalism

Post Keynesianism

Austrian school

Regulation schools

Circuitists

Feminist economics

Game theory(?)

Behavioral economics

Experimental economics

Evolutionary economics

Neuroeconomics

Complexity economics

Formal models

Atomistic, non-socially embedded individuals

Equilibrium models set out of historical time

Methodological individualism, social structures explained by aggregation over individuals

Both mainstream and non-mainstream heterodox economics can be broken down further. This can be seen in the table. The first two columns each contain more than one school of thought as an exemplar of that category. Davis makes further schematic distinctions. One is between an inward or outward orientation of heterodox economists. Another is among differents ways schools of economists can become heterodox. With these distinctions, Davis argues that the content and understanding of mainstream, non-mainstream, orthodox, and heterodox economics has been evolving over time.

Davis argues that non-mainstream economists should work harder to engage mainstream heterodox economists and that mainstream economists are more open to theoretical innovation than some non-mainstream economists claim. Without such engagement, he thinks, mainstream economists might be excessively conservative, with consequences that mainstream economists will continue to fail to incorporate worthwhile insights of non-mainstream heterodox economists. In the present historical conjuncture, I think, the odds of mainstream economics being suddenly swept away have increased. If so, Davis's strategy might be unnecessary, though I am not very optimistic either way.

By the way, I could have cited previous work by Davis for this post. I wanted to mention that Davis's is the second essay I've read in Fullbrook (2009). McFarling (2009), which is at least a stretch for me, is the first essay I read in this book. So far, I find in the little I've read in this book a broad agreement that heterodox economists reject the orthodox overemphasis on social explanations from atomistic, non-socially embedded individuals.

References

John B. Davis (2009) "The Nature of Heterodox Economics", in Fullbrook (2009)

Wednesday, April 01, 2009

Alfred Tarski's 1944 essay, "The Semantic Conception of Truth and the Foundations of Semantics" has been put up on a Web site. Tarksi's problem is to formulate a definition of truth for an object language. According to Tarski, a definition of truth is materially adequate only if condition T holds for all true sentences "p" in the object language:

(T): X is true if and only if p,

where X is the name of the sentence "p", and p is the translation of the sentence "p" into the metalanguage in which condition T is expressed.

Maybe I should explain a litte about a metalanguage and the names of sentences. The object language is merely the language for which truth of sentences is defined. It is a formal language, and some mechanism (e.g., a grammar in Backus-Naur form) is available for determining what strings are sentences in the language and what are not. Predicate calculus with quantification ("for all" and "there exists") of variables taken over natural numbers provides an example. The grammar specifies the syntax of the language, but not the semantics.

The metalanguage is merely a (formal) language in which to talk about expressions and sentences in the object language. My title provides an easy example of named sentences. "John 18:38" is the name of those sentences. Notice that "John 18:38" does not appear in the New Testament, although the verse with the name "John 18:38" does. Likewise, the names of sentences presumably do not appear in the object language.

In this informal exposition, Tarski waves his hands at a couple of points, most notably in (not) explaining what it means for the metalanguage to be "essentially richer" than the object language. The metalanguage contains translations of every sentence "p" in the metalanguage, as well as additional sentences. But this is not enough for the metalanguage to be richer. Tarski points to Bertrand Russell's theory of types and says something about the metalanguage being able to include sentences about higher types.

I'll have to study more if I really want to understand "essential richness". It occurs to me that any sentence in the metalanguage has a Gödel number. So sentences in the object language can include variables taking on a number expressing a sentence in the metalanguage. But Gödel numbering operates only on the level of syntax. I guess the idea of essential richness is to prevent the formulation of a relation like T(GN(X), GN(p)) in the object language, where this relation somehow encodes condition T and GN(X) and GN(p) are the Gödel numbers of X and p, respectively. If one could formulate a relation like this, the possibility arises of creating a sentence that says that it itself is false, under the obvious interpretation. (Footnote 11 in Section 8 of Tarski's paper provides a neat formulation of the paradox of the liar.)

I find it amazing that Tarski can relegate a proof of Gödel's incompleteness theorem to a couple of footnotes. Part of why he can do this is that he presumes all the technical machinery Gödel used to demonstrate that one can assert the provability of a sentence in the object language within the object language (given an object language in which arithmetic can be expressed). Since Tarski has shown that one cannot assert the truth of a sentence in the object language, the non-equivalence of provability and truth falls out. And all provable sentences in the object language are true. So the existence of true but unprovable sentences in the object language follows, even though Tarski, unlike Gödel, doesn't construct one.

I have some questions:

Would at least some authors of the committe that developed the Web Ontology Language be extremely well-versed in Tarski's work, including semantics and model theory? Would the same be true of at least some developers of the semantic web?

Is there a good book on model theory available and downloadable on the Web?